Rules of Exponent Manipulation

Date: Mon, 22 Jan 1996 10:22:40 -0500
From: Anonymous
Subject: Algebra
I have pushed the "I believe button" on this one but want to know the
reason behind the solution.
The problem is:
f sub r equals the reciprocal of 2 times 3.14 times square root of
0.02 times 0.18 times 10e-6
or
f sub r equals the reciprocal of 6.28 times square root of
0.0036 times 10e-6
or
f sub r equals the reciprocal of 6.28 times square root of
36 times 10e-10
or
f sub r equals the reciprocal of 6.28 times 10e-5 times square root
of 36
or
f sub r equals 10e5 divided by 37.7
What rule is it to halve the square root of 36 times 10e-10 and
move it to the left thus becoming 6.28 times 10e-5 times square root
of 36?
What rule is it to move the 10e-5 from the divisor to the dividend
making it positive?

Date: 9/1/96
From: Doctor James
Subject: Re: Algebra
This problem is easier to see when it is translated into some
internet-math shorthand. In this:
5 squared, or 5 to the power of 2 = 5^2, and 3 times 4 = 3*4.
Now, the rules of exponents say that:
(a^b)*(a^c) = a^(b+c),
[ a to the power of b times a to the power of c equals a to the
power of b plus c. See how much shorter it is without words! ]
and (a^b)^c = a^(b*c).
Also they say that
(a^c)*(b^c) = (a*b)^c.
An example of the last would be 3 to the power of 2 times 4 to the
power of 2 equals three times four to the power of 2, or
(3^2)*(4^2) = (3*4)^2 = 12^2
It is important to remember that the square root of 36 = 36^(1/2), and
the reciprocal of 10 = 1/10 = 10^(-1).
So let's write your problem in this way.
f_r = [ 2 * 3.14 * [0.02 * 0.18 * 10^(-6)]^(1/2) ]^(-1)
[what a mess!]
f_r = [ 6.28 * [ 0.0036 * 10^(-6) ]^(1/2) ]^(-1)
f_r = [ 6.28 * [ 36 * 10^(-10) ]^(1/2) ]^(-1)
[getting better!]
Now this is where we must call upon the rules of exponents. Let's
just look at the inner square brackets
[ 36 * 10^(-10) ]^(1/2)
but by the third rule I said above, this is
(36)^(1/2) * ( 10^(-10) )^(1/2)
by the second rule, this becomes
(36)^(1/2) * ( 10^[-10 * (1/2)] )
or (36)^(1/2) * ( 10^(-5) )
as -10 * (1/2) = -5. Then
f_r = [ 6.28 * 36^(1/2) * 10^(-5) ]^(-1).
The answer to your second question is similar.
f_r = [ 37.7 * 10^(-5) ]^(-1)
= ( 10^(-5) )^(-1) * 37.7^(-1)
= 10^(-5 * -1) * 37.7^(-1)
[from the second rule]
= (10^5) * 37.7^(-1)
= (10^5) / 37.7,
which is the answer you believed!
Hope this helped; if not, please email me again.
-Dr. James, The Math Forum
Check out our web site! http://mathforum.org/dr.math/